Optimal. Leaf size=142 \[ \frac {c^2 (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{30 f \sqrt {c-c \sin (e+f x)}}+\frac {c (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{20 f}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.075, Rules used = {2973, 2740, 2738} \[ \frac {c^2 (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2}}{30 f \sqrt {c-c \sin (e+f x)}}+\frac {c (5 A+B) \cos (e+f x) (a \sin (e+f x)+a)^{5/2} \sqrt {c-c \sin (e+f x)}}{20 f}-\frac {B \cos (e+f x) (a \sin (e+f x)+a)^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2738
Rule 2740
Rule 2973
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{5/2} (A+B \sin (e+f x)) (c-c \sin (e+f x))^{3/2} \, dx &=-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f}+\frac {1}{5} (5 A+B) \int (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2} \, dx\\ &=\frac {(5 A+B) c \cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{20 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f}+\frac {1}{10} ((5 A+B) c) \int (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {(5 A+B) c^2 \cos (e+f x) (a+a \sin (e+f x))^{5/2}}{30 f \sqrt {c-c \sin (e+f x)}}+\frac {(5 A+B) c \cos (e+f x) (a+a \sin (e+f x))^{5/2} \sqrt {c-c \sin (e+f x)}}{20 f}-\frac {B \cos (e+f x) (a+a \sin (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{5 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.80, size = 165, normalized size = 1.16 \[ -\frac {c (\sin (e+f x)-1) (a (\sin (e+f x)+1))^{5/2} \sqrt {c-c \sin (e+f x)} (4 (100 A+11 B) \sin (e+f x)+4 \cos (2 (e+f x)) (4 (5 A-2 B) \sin (e+f x)-15 (A+B))-3 \cos (4 (e+f x)) (5 (A+B)+4 B \sin (e+f x)))}{480 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 119, normalized size = 0.84 \[ -\frac {{\left (15 \, {\left (A + B\right )} a^{2} c \cos \left (f x + e\right )^{4} - 15 \, {\left (A + B\right )} a^{2} c + 4 \, {\left (3 \, B a^{2} c \cos \left (f x + e\right )^{4} - {\left (5 \, A + B\right )} a^{2} c \cos \left (f x + e\right )^{2} - 2 \, {\left (5 \, A + B\right )} a^{2} c\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{60 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.79, size = 147, normalized size = 1.04 \[ \frac {\left (-12 B \left (\cos ^{4}\left (f x +e \right )\right )+15 A \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+15 B \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+20 A \left (\cos ^{2}\left (f x +e \right )\right )+4 B \left (\cos ^{2}\left (f x +e \right )\right )+15 A \sin \left (f x +e \right )+15 B \sin \left (f x +e \right )+40 A +8 B \right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}}{60 f \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 16.32, size = 174, normalized size = 1.23 \[ -\frac {a^2\,c\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (60\,A\,\cos \left (e+f\,x\right )+60\,B\,\cos \left (e+f\,x\right )+75\,A\,\cos \left (3\,e+3\,f\,x\right )+15\,A\,\cos \left (5\,e+5\,f\,x\right )+75\,B\,\cos \left (3\,e+3\,f\,x\right )+15\,B\,\cos \left (5\,e+5\,f\,x\right )-400\,A\,\sin \left (2\,e+2\,f\,x\right )-40\,A\,\sin \left (4\,e+4\,f\,x\right )-50\,B\,\sin \left (2\,e+2\,f\,x\right )+16\,B\,\sin \left (4\,e+4\,f\,x\right )+6\,B\,\sin \left (6\,e+6\,f\,x\right )\right )}{480\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________